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The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh( x / a ) is the catenary , the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its probability density function is proportional to its characteristic function. However, the hyperbolic secant distribution is leptokurtic; that is, it has a more acute peak ...
Because this function can be expressed in terms of the square of the hyperbolic secant function "sech", it is sometimes referred to as the sech-square(d) distribution. [2] (See also: hyperbolic secant distribution).
The argument to the hyperbolic functions is a hyperbolic angle measure. In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant ...
Pages in category "Hyperbolic functions" ... Sech (function) Sich (mathematics) Sinh (mathematical function) Sinus hyperbolicus; Soboleva modified hyperbolic tangent; T.
The Gudermannian function relates the area of a circular sector to the area of a hyperbolic sector, via a common stereographic projection. If twice the area of the blue hyperbolic sector is ψ, then twice the area of the red circular sector is ϕ = gd ψ.
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. [1] They are named after the French mathematician Louis Poinsot . Examples of the two types of Poinsot's spirals